Deep Fréchet Regression

Advancements in modern science have led to the increasing availability of non-Euclidean data in metric spaces. This article addresses the challenge of modeling relationships between non-Euclidean responses and multivariate Euclidean predictors. We propose a flexible regression model capable of handling high-dimensional predictors without imposing parametric assumptions. Two primary challenges are addressed: the curse of dimensionality in nonparametric regression and the absence of linear structure in general metric spaces. The former is tackled using deep neural networks, while for the latter we demonstrate the feasibility of mapping the metric space where responses reside to a low-dimensional Euclidean space using manifold learning. We introduce a reverse mapping approach, employing local Fréchet regression, to map the low-dimensional manifold representations back to objects in the original metric space. We develop a theoretical framework, investigating the convergence rate of deep neural networks under dependent sub-Gaussian noise with bias. The convergence rate of the proposed regression model is then obtained by expanding the scope of local Fréchet regression to accommodate multivariate predictors in the presence of errors in predictors. Simulations and case studies show that the proposed model outperforms existing methods for non-Euclidean responses, focusing on the special cases of probability distributions and networks. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

现代科学的进步使得度量空间中非欧几里得数据的可获得性日益提升。本文旨在应对非欧几里得响应与多变量欧几里得预测因子之间关系建模的挑战。我们提出一种灵活的回归模型，无需施加参数假设即可处理高维预测因子。文中解决了两大核心挑战：非参数回归中的维度灾难，以及一般度量空间中线性结构的缺失。前者通过深度神经网络加以解决，而针对后者，我们证明了利用流形学习将响应所在的度量空间映射到低维欧几里得空间的可行性。我们引入一种反向映射方法，通过局部Fréchet回归将低维流形表示反向映射至原始度量空间中的对象。我们构建了一个理论框架，研究存在有偏依赖亚高斯噪声时深度神经网络的收敛速率。通过扩展局部Fréchet回归的适用范围以适应存在预测误差的多变量预测因子，我们得到了所提回归模型的收敛速率。仿真与案例研究表明，针对非欧几里得响应（重点关注概率分布与网络这两种特殊情形），所提模型的性能优于现有方法。本文的补充材料可在线获取，其中包含用于复现本研究的材料的标准化描述。